► Modification of the one of old method for finding exact solutions of nonlinear differential equations is considered. ► Examples of application of method are given. ► Merits and demerits of method ...are discussed.
One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and nonlinear ordinary differential equation of the seven order. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations. Merits and demerits of the method are discussed.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
The generalized Duffing oscillator Kudryashov, Nikolay A.
Communications in nonlinear science & numerical simulation,
February 2021, 2021-02-00, 20210201, Volume:
93
Journal Article
Peer reviewed
•The generalized Duffing oscillator is considered.•The Painleve test is used to study the integrability of equations.•The exact solutions in the form of periodic oscillations and solitary pulse are ...given.
A generalized Duffing oscillator is considered, which takes into account high-order derivatives and power nonlinearities. The Painlevé test is applied to study the integrability of the mathematical model. It is shown that the generalized Duffing oscillator passes the Painlevé test only in the case of the classic Duffing oscillator which is described by the second-order differential equation. However, in the general case there are the expansion of the general solution in the Laurent series with two arbitrary constants. This allows us to search for exact solutions of generalized Duffing oscillators with two arbitrary constants using the classical Duffing oscillator as the simplest equation. The algorithm of finding exact solutions is presented. Exact solutions for the generalized Duffing oscillator are found for equations of fourth, sixth, eighth and tenth order in the form of periodic oscillations and solitary pulse.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be ...considered as the model for the description of propagation pulse in a nonlinear optics. The method for finding solutions of equation is given in the general case. Solitary waves of equation are obtained as implicit function taking into account the transformation of variables.
•A new approach to analysis of the infection expansion based on first integrals of the mathematical model.•Painlevé analysis of the classic two-parameter SIR-model.•The new general analytical ...solution of the model, structurally featured through its clear relation to epidemiological data.•Re-thinking the classic SIR model with application to SARS-Cov-2 case.
A classic two-parameter epidemiological SIR-model of the coronavirus propagation is considered. The first integrals of the system of non-linear equations are obtained. The Painlevé test shows that the system of equations is not integrable in the general case. However, the general solution is obtained in quadrature as an inverse time-function. Using the first integrals of the system of equations, analytical dependencies for the number of infected patients I(t) and that of recovered patients R(t) on the number of susceptible to infection S(t) are obtained. A particular attention is paid to interrelation of I(t) and R(t) both depending on α/β, where α is the contact rate in the community and β is the intensity of recovery/decease of patients. It is demonstrated that the data on particular morbidity waves in Hubei (China), Italy, Austria, South Korea, Moscow (Russia) as well some Australian territories are satisfactorily described by the expressions obtained for I(R). The variability of parameter N having been traditionally considered as a static population size is discussed.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
•The raveling wave reduction of the mKdV hierarchy is considered.•The Lax pair associated with the traveling wave reduction of the mKdV hierarchy is given.•The first integrals of the traveling wave ...reduction for the mKdV hierarchy are found.•Exact solutions of the hierarchy and its first integrals are presented.
The traveling wave reduction of the modified Korteweg-de Vries hierarchy is considered. The linear system of equations associated with this hierarchy is found in the general form. The Lax pair is used to obtain the first integrals for the traveling wave reduction of this hierarchy. The first three members of the mKdV hierarchy are considered in more detail. Exact solutions for the traveling wave reduction for the mKdV hierarchy and its first integrals are given. The obtained first integrals can be considered as the expansion of the list for the nonlinear integrable differential equations.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé ...test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail.
Hierarchy of the perturbed nonlinear Schrödinger equations is considered. Nonlinear differential equations of this hierarchy contain higher orders and can be used for description of highly dispersive ...optical solutions. A new approach for finding solitary wave solutions of high-order nonlinear differential equations is presented. This approach allows us to significantly simplify symbolic calculations. The main idea of the method is that we use expressions of the dependent variable and its derivatives in the differential equation the polynomial form of the solitary wave. We find optical solitons with high dispersion order for nonlinear perturbed Schrodinger equations of the fourth, sixth, eighth, tenth and twelfth orders.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painlevé test we show that the generalized ...Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.
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The family of generalized Schrödinger equations is considered with the Kerr nonlinearity. The partial differential equations are not integrable by the inverse scattering transform and new solutions ...of this family are sought taking into account the traveling wave reduction. The compatibility of the overdetermined system of equations is analyzed and constraints for parameters of equations are obtained. A modification of the simplest equation method for finding embedded solitons is presented. A block diagram for finding a solution to the nonlinear ordinary differential equation is given. The theorem on the existence of bright solitons for differential equations of any order with Kerr nonlinearity of the family considered is proved. Exact solutions of embedded solitons described by fourth-, sixth-, eighth and tenth-order equations are found using the modified algorithm of the simplest equation method. New solutions for embedded solitons of generalized nonlinear Schrödinger equations with several extremes are obtained.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
A method for finding solitary wave solutions to nonlinear differential equations is presented. A generalization for the logistic function to obtain a solitary wave solution is introduced. Properties ...of this basic function are discussed. An algorithm for finding exact solutions in the form of a solitary wave for nonlinear differential equations is formulated. The method has significant advantages over other approaches of this type. Its advantage is due to the fact that in the calculations we do not use the form of a specific function. Our approach is particularly effective in finding exact solutions to high-order nonlinear differential equations used in describing the propagation of pulses in an optical fiber. The application of the exact solutions search method for finding highly dispersed solitons of nonlinear differential equations is shown.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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